Area of an Ellipse

Figure 1

You know the formula for the area of a circle of radius R.
It is Pi*R2.

But what about the formula for the area of an ellipse of
semi-major axis of length A and
semi-minor axis of length B? (These semi-major axes are half the lengths of, respectively, the largest and smallest diameters of the ellipse--- see Figure 1.)

For example, the following is a standard equation for such an ellipse centered at the origin:

(x2/A2) + (y2/B2) = 1.

The area of such an ellipse is

Area = Pi * A * B ,

a very natural generalization
of the formula for a circle!

Presentation Suggestions:
If students guess this fact, ask them what they think
the volume of an ellipsoid is!

The Math Behind the Fact:
One way to see why the formula is true is to realize
that the above ellipse is just a unit circle that has been
stretched by a factor A in the x-direction, and a factor
B in the y-direction. Hence the area of the ellipse
is just A*B times the area of the unit circle.

The formula can also be proved
using a trigonometric substitution.
For a more interesting proof, use line integrals
and Green's Theorem in multivariable calculus.

Each of the above proofs will generalize to
show that the volume of an ellipsoid with
semi-axes A, B, and C is just

(4/3)*Pi*A*B*C.

(Just think of a stretched sphere, use trig substitution, or use
an appropriate flux integral.)

By the way, unlike areas, the formula for the length of the perimeter of a circle does not generalize in any nice way to the perimeter of an ellipse, whose arclength is not expressible in closed form--- this difficulty gave rise to the study of the so-called elliptic integrals.